Differentiation

In calculus, differentiation is one of the two important concept apart from integration. It is a method of finding the derivative of a function or instantaneous rate of change in function based on one of its variables. If x is a variable and y is another variable, then the rate of change of x with respect to y is given by dy/dx.

Functions are generally classified in two categories under Calculus, namely:

(i) Linear functions

(ii) Non-linear functions

A linear function varies with a constant rate through its domain. Therefore, the overall rate of change of the function is the same as the rate of change of a function at any point.

However, the rate of change of function varies from point to point in case of non-linear functions. The nature of variation is based on the nature of the function.

The rate of change of a function at a particular point is defined as a derivative of that particular function.

Differentiation in Calculus

Differentiation, in terms of calculus, can be defined as a derivative of a function regarding the independent variable and can be applied to measure the function per unit change in the independent variable.

Let y = f(x) be a function of x. Then, the rate of change of “y” per unit change in “x” is given by:

dy / dx

If the function f(x) undergoes an infinitesimal change of h near to any point x, then the derivative of the function is defined as

\(\lim\limits_{h \to 0} \frac{f(x+h) – f(x)}{h}\)

When a function is denoted as y=f(x), the derivative is indicated by the following notations.

  1. D(y) or D[f(x)] is called Euler’s notation.
  2. dy/dx is called Leibniz’s notation.
  3. F’(x) is called Lagrange’s notation.

Differentiation is the process of determining the derivative of a function at any point.

Differentiation Formulas

Some of the important Differentiation formulas in differentiation are as follows.

  1. If f(x) = tan (x), then f'(x) = sec2x
  2. If f(x) = cos (x), then f'(x) = -sin x
  3. If f(x) = sin (x), then f'(x) = cos x
  4. If f(x) = ln(x), then f'(x) = 1/x
  5. If f(x) = \(e^{x}\), then f'(x) = \(e^{x}\)
  6. If f(x) = \(x^{n}\), where n is any fraction or integer, then f'(x) = \(nx^{n-1}\)
  7. If f(x) = k, where k is a constant, then f'(x) = 0

Also, see:

Differentiation Rules

Some of the basic differentiation rules that need to be followed are as follows.

(i) Sum or Difference Rule

If the function is sum or difference of two functions, the derivative of the functions is the sum or difference of the individual functions, i.e.,

If f(x)=u(x)±v(x)

then, f'(x)=u'(x)±v'(x)

(ii) Product Rule

As per the product rule, if the function f(x) is product of two functions u(x) and v(x), the derivative of the function is,

If \(f(x) = u(x) \times v(x)\)

then, \(\mathbf { f'(x) = u'(x) \times v(x) + u(x) \times v'(x)}\)

(iii) Quotient rule

If the function f(x) is in the form of two functions [u(x)]/[v(x)], the derivative of the function is

If, \(f(x) = \frac{u(x)}{v(x)}\)

then, \(\large \mathbf { f'(x) = \frac{u'(x) \times v(x) – u(x) \times v'(x)}{(v(x))^{2}}}\)

(iv) Chain Rule

If a function y = f(x) = g(u) and if u = h(x), then the chain rule for differentiation is defined as,

\(\large \mathbf{\frac{\mathrm{d} y}{\mathrm{d} x} = \frac{\mathrm{d} y}{\mathrm{d} u} \times \frac{\mathrm{d}u }{\mathrm{d} x}}\)

This plays a major role in the method of substitution that helps to perform differentiation of composite functions.

Differentiation Examples

With the help of differentiation, we are able to find the rate of change of one quantity with respect to another. Some of the examples are:

  • Acceleration: Rate of change of velocity with respect to time
  • To calculate the highest and lowest point of the curve in a graph or to know its turning point, the derivative function is used
  • To find tangent and normal to a curve

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